Thursday, September 11, 2008
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Today, dozens of mostly British media including Nature, The Telegraph, The (London) Times, Register, First Post, and many others have exaggerated a story about a "battle of titans": Prof Peter Higgs "attacked" Prof Stephen Hawking, they argue. ;-)

What actually happened was that Stephen Hawking wanted to be interesting, so he "conjectured" that the LHC won't find the Higgs boson. Instead, it could find SUSY or something completely different (or unfamiliar), Hawking claimed. The money he bet, USD 100, are unlikely to bring Stephen Hawking close to bankruptcy. ;-)

Prof Higgs was irritated a bit and began to talk about Hawking's paper that is "not good enough". Well, I don't think that there exists any real technical paper by Hawking that supports his no-Higgs scenario. Strange things can happen but the papers that argue that there is nothing like the Higgs boson at all have so far been crappy so I would say that Peter Higgs was correct - and excessively diplomatic because those papers are not really "not good enough" but rather piles of sh*t - and he was correct even if he was talking about a non-existent paper by Hawking.

It sucks, anyway. ;-) Stephen Hawking is a revolutionary physicist but if he wrote a paper claiming that nothing like the Higgs sector exists, such a paper would really be "not good enough", to use the diplomatic jargon. The journalists clearly misunderstand that science doesn't mean that some people's work becomes uncriticizable once they repeatedly appear in the media. Every scientist can write a wrong paper and almost all of them have. And Hawking's Higgs-less paper doesn't even exist. ;-)

Update: A commenter has pointed out that the relevant paper actually does exist and was written in 1995: Virtual Black Holes. The paper talks about some S2 x S2, CP2, and K3 quantum gravity (!) instantons. These instantons may perhaps be interesting (not sure why he doesn't talk about del Pezzo and other shapes) but I don't think that they can possibly have anything to do with physics at the LHC scale.

In this paper, Hawking doesn't seem to be bothered by the difference between a TeV and the Planck scale at all and the acronyms "GeV" or "TeV" don't even appear in the paper which means that the accuracy of his analysis is too rough to address any of the counter-proofs discussed below. So my comments about this paper hold even when I saw it. That doesn't mean that I don't consider Hawking to be an amazing physicist. But this paper is just crap.

If the appearance of quantum gravity at the electroweak scale was not bad enough for you, Hawking also includes another favorite meme of that time, a loss of unitarity. Among low-energy effects, these non-unitary effects are supposed to influence scalar fields only (and solve the strong CP-problem, too) because of reasons that look manifestly wrong to me (if he were right, we shouldn't observe other low-energy spin zero particles either, for example para-hydrogen). I thought that he has already abandoned these non-unitary things!

Tree unitarity

How do we know that something like the Higgs boson must exist? Consider some process of elastic scattering. We could talk about quarks or gluons or leptons (that usually collide at the colliders) but because this argument is theoretical in character, we can choose any particles we like as long as they are already known to exist.

So let's scatter one W+ boson with one W- boson in the initial state and the same pair in the final state. These particles have already been observed. They exist so it must also be possible to scatter them.

For technical reasons, it is especially useful to consider particular polarizations of the spin. These W bosons are known to have spin 1. As all massive particles with spin 1, their wave functions have 3 complex components (for m=-1,0,+1, if you wish). These components are in one-to-one correspondence with a complex vector in 3 dimensions.

Because of rotational symmetry, the particle can be polarized with respect to any axis. Let's take both W bosons in the initial state and the final state to be longitudinally polarized - which means that the projection of their spin to the direction that connects the two W bosons is zero (instead of -1 or +1). For massive spin 1 bosons, the longitudinal polarization is as physical as others. In gauge theory, you know that it comes from "eaten" components of the Higgs multiplet but you can pretend that you don't know about this origin of the polarization.

Now, what is the cross section of such scattering? The first relevant Feynman diagram comes from a quartic vertex - four W bosons meet at one point. And then there are Feynman diagrams with a photon and a Z boson in the intermediate state (of an s-channel). These diagrams must contribute because quantum field theory with these known particles (photon, W, Z) works, at least at low enough energies.

If you only sum the diagrams above, you get a cross section - or, more conveniently, the scattering amplitude - that doesn't behave well enough at fixed angles, high energies (that are scaled to infinity). In fact, you could prove that the total probability that similar reaction occurs could exceed 100%: the cross section would increase too rapidly with the energy, as a power law with too high an exponent.

Needless to say, Nature knows how to fix this thing. There must be another Feynman diagram i.e. another contribution to the elastic scattering process. Instead of a photon or a Z boson, it has a Higgs boson (so far undiscovered particle) in the intermediate state. This contribution only becomes significant when the energy of the W bosons approaches the mass of the Higgs boson. But when it is so, the leading behavior of the new Higgs-induced term at high energies is exactly correct so that it cancels the most divergent part of the previous diagrams that only included the observed particles.

The full sum, including the God particle in the intermediate state, behaves much more nicely at high energies, as can be easily checked. But the paper by Cornwall et al. actually implies that the opposite implication is also true, given some assumptions: the only way how to make the massive bosons interact "nicely" at high energies, so that the limit 100% for probabilities is never exceeded, is to generate their mass from an interaction with a Higgs boson.

Now, you could try to look for loopholes. There could be large loop corrections. Maybe: except that the electroweak coupling constants are known to be pretty small near the 1 TeV scale and they shouldn't matter much (if the loops were able to cancel the tree diagrams, you would also be close to the point where the theory diverges and collapses, anyway).

You could also think that the new diagrams could be much more complicated than the exchange of a single scalar particle. Something composite and "non-perturbative" could be exchanged instead. Maybe: except that this new object must "effectively" look like a scalar Higgs boson at the TeV energies, as can be proved.

Perhaps, there can be two Higgses or infinitely many new things that are exchanged in between the W bosons. If you chose this comment, you would be entirely correct. The supersymmetric extension of the Standard Model, or the MSSM, indeed requires two Higgs doublets. And in theories with very large or warped dimensions, you could get infinitely many new players - especially the Kaluza-Klein modes of known particles.

If you look at all the possibilities, you will always see that there must exist something new that is being exchanged: the Standard Model with the known particles and the known interactions is simply inconsistent because the probabilities can exceed 100%. And this new object must look like at least one scalar particle at the TeV energies. Such a new particle - with the new diagrams - must enter the scene early enough to prevent the probabilities from exceeding 100%.

A priori, you could think that the new possible physical mechanisms occupy a wonderland full of dragons. But if you ask the right question - e.g. what is the cross section of the WW elastic scattering - and study the cross section carefully, with all expansions and other mathematical tricks you have learned, it is actually possible to get rid of the dragon nightmares. There are no dragons but the Higgs boson must be there.

This reasoning implies that at least the lightest Higgs-like particle among those that contribute must be lighter than something like 800 GeV: the theory where the lightest Higgs-like particle is as heavy as possible is actually the normal Standard Model. Other theories with many Higgses, such as supersymmetric models, typically require the lightest Higgs boson to be lighter or much lighter than 800 GeV.

To summarize, something that behaves almost indistinguishably from the usual point-like Higgs boson must exist to restore the unitarity of the WW elastic scattering. In principle, it may be composite but such a possibility - compositeness - is not unrestricted. There are all kinds of constraints that show that the compositeness must be pretty much undetectable by the LHC.

We will see what the right answer is but as long as the LHC will operate for many years, Stephen Hawking's provocative yet wishful thinking (his dreams to create a complete confusion in theoretical particle physics) will almost certainly be proved incorrect.